3.253 \(\int \frac{x^3}{\sqrt{a x^2+b x^3}} \, dx\)

Optimal. Leaf size=75 \[ \frac{16 a^2 \sqrt{a x^2+b x^3}}{15 b^3 x}-\frac{8 a \sqrt{a x^2+b x^3}}{15 b^2}+\frac{2 x \sqrt{a x^2+b x^3}}{5 b} \]

[Out]

(-8*a*Sqrt[a*x^2 + b*x^3])/(15*b^2) + (16*a^2*Sqrt[a*x^2 + b*x^3])/(15*b^3*x) +
(2*x*Sqrt[a*x^2 + b*x^3])/(5*b)

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Rubi [A]  time = 0.160626, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 a^2 \sqrt{a x^2+b x^3}}{15 b^3 x}-\frac{8 a \sqrt{a x^2+b x^3}}{15 b^2}+\frac{2 x \sqrt{a x^2+b x^3}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a*x^2 + b*x^3],x]

[Out]

(-8*a*Sqrt[a*x^2 + b*x^3])/(15*b^2) + (16*a^2*Sqrt[a*x^2 + b*x^3])/(15*b^3*x) +
(2*x*Sqrt[a*x^2 + b*x^3])/(5*b)

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Rubi in Sympy [A]  time = 19.0388, size = 66, normalized size = 0.88 \[ \frac{16 a^{2} \sqrt{a x^{2} + b x^{3}}}{15 b^{3} x} - \frac{8 a \sqrt{a x^{2} + b x^{3}}}{15 b^{2}} + \frac{2 x \sqrt{a x^{2} + b x^{3}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**3+a*x**2)**(1/2),x)

[Out]

16*a**2*sqrt(a*x**2 + b*x**3)/(15*b**3*x) - 8*a*sqrt(a*x**2 + b*x**3)/(15*b**2)
+ 2*x*sqrt(a*x**2 + b*x**3)/(5*b)

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Mathematica [A]  time = 0.0259506, size = 42, normalized size = 0.56 \[ \frac{2 \sqrt{x^2 (a+b x)} \left (8 a^2-4 a b x+3 b^2 x^2\right )}{15 b^3 x} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[x^2*(a + b*x)]*(8*a^2 - 4*a*b*x + 3*b^2*x^2))/(15*b^3*x)

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Maple [A]  time = 0.009, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 3\,{b}^{2}{x}^{2}-4\,abx+8\,{a}^{2} \right ) x}{15\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^3+a*x^2)^(1/2),x)

[Out]

2/15*(b*x+a)*(3*b^2*x^2-4*a*b*x+8*a^2)*x/b^3/(b*x^3+a*x^2)^(1/2)

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Maxima [A]  time = 1.39806, size = 57, normalized size = 0.76 \[ \frac{2 \,{\left (3 \, b^{3} x^{3} - a b^{2} x^{2} + 4 \, a^{2} b x + 8 \, a^{3}\right )}}{15 \, \sqrt{b x + a} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x^3 + a*x^2),x, algorithm="maxima")

[Out]

2/15*(3*b^3*x^3 - a*b^2*x^2 + 4*a^2*b*x + 8*a^3)/(sqrt(b*x + a)*b^3)

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Fricas [A]  time = 0.218333, size = 54, normalized size = 0.72 \[ \frac{2 \,{\left (3 \, b^{2} x^{2} - 4 \, a b x + 8 \, a^{2}\right )} \sqrt{b x^{3} + a x^{2}}}{15 \, b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x^3 + a*x^2),x, algorithm="fricas")

[Out]

2/15*(3*b^2*x^2 - 4*a*b*x + 8*a^2)*sqrt(b*x^3 + a*x^2)/(b^3*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{x^{2} \left (a + b x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**3+a*x**2)**(1/2),x)

[Out]

Integral(x**3/sqrt(x**2*(a + b*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{b x^{3} + a x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x^3 + a*x^2),x, algorithm="giac")

[Out]

integrate(x^3/sqrt(b*x^3 + a*x^2), x)